Dans cet exposé j'essayerai de donner un panorama d'inégalités isopérimétriques démontrées ou conjecturées, en donnant des idées de preuves ou des difficultés. Dans ce cadre je présenterai un résultat positif récent obtenu en collaboration avec Greg Kuperberg (UC Davis), concernant le profil isopérimétrique des variétés de courbure majorée.

The question of Plateau concerns the existences soap films: objects of minimial area spanning a given curve in the Euclidean spaces. The most classical answer to this question has been provided by Rado and Douglas and proves the existence of parametrized discs of minimal area spanning an arbitrary Jordan curve. The result was generalized by Morrey to Riemannian manifolds. In the talk I will discuss a solution of the Plateau problem in arbitrary metric spaces, regularity of the solutions and some applications to isoperimetric problems.

A Riemannian manifold is called static when it admits a non-trivial solution to a certain equation that appears both in Geometry (e.g, in the problem of prescribing the scalar curvature) and in Physics (e.g., in the study of static space-times). The staticity condition is very restrictive, specially in dimension three. We will discuss some classification results in the positive scalar curvature case.

We show area growth estimates for minimal graphs in the Heisenberg space.

We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire minimal graphs in the Heisenberg space have at most cubic intrinsic area growth.

Moreover we explain why height estimates of graphs are useful in the comprehension of the area growth and we show a Collin-Krust type result.

Some of our results can be extended to graphs with constant critical mean curvature in homogeneous spaces.

Joint work with José Miguel Manzano.

Soit $G$ un graphe plongé dans la sphère. Quand est-ce que $G$ est le $1$-squelette d'un polyèdre inscrit dans un hyperboloïde à une nappe ?

On montrera que c'est le cas si et seulement si $G$ est le $1$-squelette d'un polyèdre inscrit dans la sphère et qu'il a un cycle hamiltonien.

La preuve repose sur la description des angles dièdres des polyèdres idéaux dans l'espace anti-de Sitter. Un résultat analogue s'applique aux polyèdres inscrits dans un cylindre, en relation avec une géométrie "transitionnelle" entre hyperbolique et anti-de Sitter.

Travail en commun avec Jeff Danciger et Sara Maloni.

In this talk we investigate the geometry and topology of $f$-extremal domains in a manifold with negative sectional curvature. A $f$-extremal domain is a domain that supports a positive solution to the overdetermined elliptic problem
\[
\begin{cases}
\Delta u+f(u)=0&\textrm{in }\Omega,\\
u>0&\textrm{in }\Omega,\\
u=0&\textrm{on }\partial\Omega,\\
\langle\nabla u,\vec\nu\rangle_M=\alpha&\textrm{on }\partial\Omega,
\end{cases}
\]
where $\Omega$ is an open connected domain in a complete Hadamard $n$-manifold $(M, g)$ with boundary $\partial\Omega$ of class $C^2$ , $f$ is a given Lipschitz function, $\langle\cdot,\cdot\rangle_M$ is the inner product on $M$ induced by the metric $g$, $\vec\nu$ the unit outward normal vector of the boundary $\partial\Omega$ and $\alpha$ a non-positive constant.

We will show narrow properties of such domains in a Hadamard manifolds and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity. Later, we focus on $f$−extremal domains in the Hyperbolic Space $\mathbb{H}^n$ . Symmetry and boundedness properties will be shown. Hence, we are able to prove the Berestycki-Caffarelli-Nirenberg Conjecture in $\mathbb{H}^2$ . Specifically:

$\textbf{Theorem:}$ Let $\Omega\subset\mathbb{H}^2$ a domain with properly embbeded $C^2$ connected boundary such that $\mathbb{H}^2\setminus\Omega$ is connected. If there exists a (strictly) positive function $u\in C^2(\Omega)$ that solves the equation
\[
\begin{cases}
\Delta u+f(u)=0&\textrm{in }\Omega,\\
u>0&\textrm{in }\Omega,\\
u=0&\textrm{on }\partial\Omega,\\
\langle\nabla u,\vec\nu\rangle_M=\alpha&\textrm{on }\partial\Omega,
\end{cases}
\]
where $f : (0, +\infty) \rightarrow \mathbb{R}$ satisfies $f (t)\ge\lambda t$ for some constant $\lambda$ satisfying $\lambda > 1/4$, then $\Omega$ must be a geodesic ball and $u$ radially symmetric.

If time permits, we will generalize the above results to more general OEPs.